Download First Semester Final Exam Review Part I: Proofs 1. Given: AB ≅ BC 2.

Honors Geometry
First Semester Final Exam Review
Part I: Proofs
1. Given: AB @ BC
AE @ EC
Prove: AD @ DC
Name: _____________________
2.
Given:
Prove:
3.
Given:
Prove:
5.
Given:
Prove:
7.
ZY bisects SX
ST @ WX
SX bisects ZY
RSZW @ RXYT
4.
RSOT is a parallelogram
MS @ TP
MOPR is a parallelogram
Given: NPRS is a parallelogram
with diagonals SP and NR intersect @ O.
TO ^ plane of parallelogram NPRS
Prove: ΔSTP is isosceles
Given:
Prove:
ABDE is a parallelogram
BC is the base of isosceles ΔBCD
ACDE is an isosceles trapezoid
6.
Given:
Prove:
TVAX is a rectangle
RTXV @ RVAT
8.
Given:
Prove:
AB @ AC, BD @ DC
RADB @ RADC
9.
Given:
GH @ GK
HM @ KM
HMKJ is a kite
Prove:
10. Given:
Prove:
AD @ BC
RDAB @ RCBA
ΔABE is isosceles
11. Given:
Prove:
12. Given:
Prove:
AC @ AD , FC @ DG
R1 @ R2
BE ║ CD
FCDG is an
isosceles trapezoid
RRHG @ RKHJ
OH is median to GJ
OH is altitude to GJ
RP @ RM
FJ is the base of an isosceles ∆
FG @ JH, O is the midpoint of MF
K is the midpoint of MJ
OH @ KG
13. Given:
ΔKOR is equilateral
KOPR is a parallelogram
KMOR is a parallelogram
Prove: ΔJMP is equilateral
1
14. Given:
BA ^ AC
15. Given:
Prove:
Prove:
DC ^ AC
DC @ BA
RB @ RD
16. Given:
AB @ CB
17. Given:
Prove:
BD ^ AC
D is not the
midpoint of AC
Prove:
RA @ RD
C bisects BE
AB @ ED
R1 @ R2
R3 @ R4
ΔABE @ ΔDCE
Part II: Final Exam Review. You may need to draw the diagram to solve the problem.
1. Find x.
2. Answer always, sometimes, or never.
a) If a triangle is obtuse it is isosceles.
b) The bisector of the vertex angle of a scalene ∆ is perpendicular to the base.
c) If one of the diagonals of a quadrilateral is the perpendicular bisector of the
other the quadrilateral is a kite.
d) If A, B, C and D are non-coplanar, AB ^ BC, and AB ^ BD, then AB is
perpendicular to the plane determined by B, C, and D.
e) Two parallel lines determine a plane.
f) Planes that contain two skew lines are parallel.
g) Supplements of complementary angles are congruent.
3. FGHJ is a parallelogram, FG = x + 5, GH = 2x + 3, mRG = 40°, mRJ = 4x + 12. Find: mRF, perimeter of FGHJ
4. ABCD is a parallelogram, mRA = 3x + y, mRD = 5x + 10, mRC = 5y + 20. Find mRB.
5. The measure of the supplement of an angle exceeds three times the measure of the complement of the
angle by 12°. Find the measure of half of the supplement.
6. Write the most descriptive name for each figure:
a) A four-sided figure in which the diagonals are perpendicular bisectors of each other.
b) A four-sided figure in which the diagonals bisect each other.
c) A triangle in which there is a hypotenuse.
d) A four-sided figure in which the diagonals are @ and all sides are @ .
7. If one of two supplementary angles is 16° less than three times the other find the measure of the larger.
8. Two consecutive angles of a parallelogram are in the ratio of 7 to 5. Find the measure of the larger.
9. Find mR1 if a ║ b.
10. Given: ΔFJH is isosceles with base JH, K and G are
midpoints, FK = 2x + 3, GH = 5x – 9, JH = 4x
Find:
The perimeter of ΔFHJ
2
11. a) How many points determine a line?
c) Collinear means?
b) How many non-collinear points determine a plane?
d) Coplanar means?
12. Fill in each blank with line, segment, or ray.
a) A _____________________ has one endpoint. b) A _____________________ has a definite length.
c) A _____________________ can be bisected. d) A _____________________ has two endpoints.
e) A _____________________ has no endpoints. f) A _____________________ has no midpoints.
g) The union of two opposite rays is called a _____________________.
13. SR = RQ = QT and T is the midpoint of SM.
a) If SR = 20, then RT = _______
b) If TM = 45, then RQ = _______
c) If QT = 12, then QM = _______
d) If RT = 8, then SM = _______
14. Use the figure at the right.
a) m Ð DQM =
b) m Ð DQY =
c) m Ð TQM =
d) m Ð TQS =
e) m Ð SQY =
f) m Ð DQT =
g) Name two right angles.
h) Name two obtuse angles that have QD as a side.
i) Name three acute angles that have QM as a side.
j) __________ bisects Ð MQS.
15. Name the following parts of isosceles ∆ABC.
a) legs
b) vertex angle
c) base angles
Q
16. Name the following parts for right ∆TMR.
a) right angle
b) hypotenuse
c) legs
17. Classify each triangle according to its angle measures:
a) 17o, 80o, 83o
b) 45o, 45o, 90o c) 60o, 60o, 60o
d) 25o, 65o, 90o
e) 10o, 10o, 160o
18. Use the figure at the right for the following true/false statements.
sur
suur
a) AC and DE intersect in B.
b) E, B, F determine a plane.
c) The plane determined by C, B, D contains E.
d) C and B determine a plane.
e) Only one plane contains D, B, E.
sur
f) Only one plane contains DB and C.
g) C, B, H, and E are coplanar.
h) C, B, E, and D are coplanar.
sur
i) The intersection of planes m and n is BA .
3
20. If mRR = 2x + 7 and the measure of the supplement of RR = x + 8, find mRR.
19. Use the diagram to find:
a) Ð EIB
b) Ð BIC
c) Ð HIB
21. If mRNOM = 37 and mRMOP = 73 find mRNOP.
I
22. In this figure how many angles are adjacent to RRST ?
23. Find the measure of the supplement of RA if it is five times the measure of RA.
24. If the measure of the complement of the angle is 10 less than ½ the measure of the supplement of the
same angle find the measure of the angle, its complement, and its supplement.
25. Find the complement and the supplement of a 26o17’ angle.
uur uur
uur
uur
uur
uur
26. If EF , EG , and EH are coplanar and EF is between EG and EH , then m R FEG + m R ______ = m R ______.
27. Complete each statement:
a) A triangle which has three congruent sides is an _________ triangle.
b) The _________ of a statement says the opposite of the original statement.
c) A triangle which has two congruent sides is an _________ triangle.
d) Two triangles are congruent if their _________ are congruent.
e) A geometric figure is congruent to itself by the _________ property.
f) If two sides of a triangle are congruent then the _________ opposite these sides are @ .
g) Every equiangular triangle is an _________ triangle.
h) A _________ always contains the phrase if and only if.
i) Any point on the _________ of a segment is equidistant from the endpoints of the segment.
28. Write the reason each pair of triangles is congruent. SAS, ASA, SSS, AAS, HL, or none.
a)
b)
c)
d)
e)
f)
29. Find the missing angles and, where possible, the missing sides.
a)
b)
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30. Classify each statement as true or false.
a) If R1 ≅ R2, then BA @ BC .
b) If RBCA ≅ RBAC, then BA @ BC
c) If BA = BC, then BD is the perpendicular bisector of AC .
d) If EA = EC, then BD is the perpendicular bisector of AC .
e) If BA = BC and EA = EC, then B, E, and D, are collinear.
31. Given: KM║NO
Find: m R N
32. Given: MN║PO
Find: all pairs of @ R' s .
33. If R4 ≅ R9, then _____ ║ _____
36. A pair of interior angles on the same side of
34. If a║b, then mR1 =
the transversal is:
35. If a║b, then mR3 =
37. A pair of corresponding angles is:
38. A pair of alternate interior angles is:
39. If DE ≅DF, mRF =
40. Find x and mRD.
41. R1 ≅ R2, mRB = 100 o, and mRC = 35 o.
42. Find x and mRA.
Find:
mR1,
mR2,
mR3,
mR4, and
mRA.
43. If AE║BD, BF║CE,
44.
l
║XY, mR2 =
mR1 = 100 o,
then mRDBF =
5
45. Given: a║b, x =
46. If KM║NO, which angles are supplementary?
47. mR1 = mR_____ + mR _____
48. R2 is an exterior angle of which triangle?
49. If PQ║ST, mRP = 48 o, mRPRQ = 110o,
50. l ║ m , find mR1, mR2, mR3, mR4, mR5.
find mRQ, mRS, mRT.
51. If REFG is a right angle, FH^HG,
o
mR2 = 40 , then mRE =
52. If l ║ m , mR2 = 5x-9, mR6 = 2x+18,
then find x and mR2.
53. S, T, and W are the midpoints of the sides of this triangle.
a) If DE = 18, then WT =
b) If WS = 11, then 22 =
c) If WT = 5, ST = 8, SW = 7, then the perimeter of ΔDEF =
54. ABCD is a square.
Fill in all angle measures.
55. EFGH is a rhombus.
Fill in all angle measures.
56. JKLM is a rectangle.
Fill in all angle measures.
57. NOPQ is a parallelogram.
Fill in all angle measures.
58. RSTU is an isosceles trapezoid.
Fill in all angle measures.
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59. A, B, C, and D are midpoints of VY,VW,WX, and XY.
If VX = 20 and CD = 3, find BA, WY, AD, and BC.
60. If two sides of a triangle have measures 5 and 12,
then the third side is between _____ and _____ .
61. If mRA > mRB > mRC in triangle ABC, which side is largest?
62. Parallelogram ABCD. If mRA = 5x-20 and mRC = 3x, find x and mRB.
63. If the perimeter of the parallelogram in #62 is 120 and AB = 4x+20 and BC = 6x-10, find x, AB, and AD.
64. The coordinates of A and B are (2,7) and (-3,5), respectively. Find the coordinates of the midpoint of AB.
Find the slope of AB.
65. Show using slope that ABC is a right triangle if A(4,6), B(1,2), and C(5,-1).
66. If M is the midpoint of AB and A(-3,-6) and M(7,9), find the coordinates of B.
67. An exterior angle of a regular convex polygon is 15o. How many sides does the polygon have?
68. What is the sum of the measures of the angles of a convex hexagon? If three of the angles are
120o, a fourth is 84o, and the fifth is twice the sixth, find the measure of the sixth angle.
69. Find the number of diagonals of a convex dodecagon.
70. Find the measure of each interior angle of a regular 32-gon.
71. Find the number of sides of a regular polygon if the measure of each interior angle is eleven times the
measure of each exterior angle.
72. Write the converse, inverse, and contrapositive of: If it is warm today, then H-F will win.
73. A(2,6), B(8,-2), and C(-10,4).
Find the slope of the median to side:
a) AB
b) BC
c) AC
74. Use the same points and find the slope
of the altitude to side:
a) AB
b) BC
c) AC
75. If M(-7,2) is the midpoint between A(8,1) and B, find the coordinates of B.
76. Prove ABCD is a rectangle if A(-2,3), B(8,3), C(8,1), and D(-2,1).
77. Find the slope of AC so that mRBCA is a right angle if B(-2,4) and C(6,1).
78. Given: KITE is a kite and KT is the ^ bisector of EI. KI = 6x + 2y – 2, IT = 2x + 3y
TE = 4x + 2y – 3, KE = 3x + 4y.
Find: x, y, and the perimeter of KITE
79. Given:
PQR @ STV. PQ = x2, SV = 6, ST = 2x + 15, TV = 3 – x.
Find : a) All possible x values.
b) The perimeter of PQR.
c) Is PQR scalene, isosceles, or equilateral
V
V
V
V
80. What conclusion can be drawn from the following: : c Þ: f , g Þ b, p Þ f , c Þ: b
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Answers
1) 1. given 2. Draw AC; 2 pts det line 3. BD ^bis AC; If 2 pts on a line are =dist from seg endpts, then ^ bis.
4. AD ≅ AC; If a pt lies on ^bis, then it is = dist from seg endpts.
2) 1. given 2. OH ^ GJ; If altitude, then ^ 3. R OHJ & ROHG Rt R’s; If ^, then Rt. 4. ROHJ ≅ROHG; If Rt R, then ≅
5. OH ≅ OH; Reflexive 6. GH ≅ JH; if median, then div into 2 ≅ seg 7. ΔGHO ≅ ΔJHO; SAS 8. RG ≅ RJ; CPCTC
9. RPHO ≅ RMHO; If 2 ≅ R;s sub from 2 ≅ R’s then diff ≅
10. RGHM ≅RJHP; ; If 2 ≅ R;s added to 2 ≅ R’s then sums ≅
11. ΔGHM ≅ ΔJHP; ASA 12. RP ≅ RM; CPCTC
3) 1. Given 2. RS ≅ RX & RZ ≅ RY; If bis, then 2 ≅ seg 3. RT ≅ RW; If 2 ≅ seg sub from 2 ≅ seg, then diff ≅
4. RZRW ≅ RYRT & RZRS ≅ RYRX; Vert R’s ≅ 5. ΔTRY ≅ Δ ZRW; SAS 6. RRTY ≅ RRWZ; CPCTC
7. ΔXRY ≅ ΔSRZ; SAS 8. RS ≅ RX; CPCTC 9. RSZW ≅ RXYT; No choice thm
4) 1. Given 2. ED ║ AB; If ║ogram, then opp sides ║ 3. ACDE is a trap; If 1 pr opp sides ║ then trap
4. AE ≅ BD; If ║ogram, then opp sides ≅ 5. BD ≅ DC; If isos Δ, then legs ≅ 6. AE ≅ DC; transitive
7. ACDE is isos trap, If trap has 2 ≅ legs, then isos.
5) 1. Given 2. RS ≅ TO; If ║ogram, then opp sides ≅ 3. RS ║ TO; If ║ogram, then opp sides║
4. RRSM ≅ RPTO; If ║ lines, then alt ext R’s ≅ 5. ΔRSM ≅ ΔOTP; SAS 6. MR ≅ PO; CPCTC 7. RRMS ≅ ROPT ; CPCTC
8. MR ║ PO; If 1 pr opp sides ║ & ≅ then ║ogram
6) 1. given 2. XT ≅ VA; If rect, then opp sdies ≅ 3. TV ≅ TV; reflexive 4. XV ≅ TA; If rect, then diag ≅
5. ΔXTV ≅ ΔAVT; SSS 6. RTXV ≅ RVAT; CPCTC
7) 1. given
2. TO ≅ TO; Reflexive
3. TO ^ OP, OS; If ^ plane, then ^ all lines in plane thru foot
4. RTOP & RTOS Rt R’s; If ^, the Rt R’s
5. RTOP ≅ RTOS; If rt R’s, then ≅
6. SO ≅ PO; If parallelogram, then diag bis each other
7. ΔTOS ≅ ΔTOP; SAS
8. TS ≅ TP; CPCTC
9. ΔSTP is isos; If 2 sides ≅, then isos.
8) 1. list 2 possibilities
2. RADB ≅ RADC; assume one poss. to be true 3. Draw AD; 2 pts det a line
4. AD ≅ AD; reflexive
5. ΔADB ≅
ΔADC; SAS , 6. AB ≅ AC; CPCTC 7. AB not ≅ AC, given 8. Assumption was false, contradiction 9. RADB not ≅ RADC , only remaining poss.
9) 1. Given 2. Draw HK; 2 pts det a line
3. GJ ^bis HK; If 2 pts on a line are =dist from seg endpts, then line is ^bis to the seg
4. HMKJ is a kite; If 1 diag ^bis to other diag, then kite.
10) 1. Given
2. AB ≅ AB; Reflexive
3. ΔDAB ≅ ΔCBA; SAS
4. RDBA ≅ CAB; CPCTC
5. ΔABE is isos; If 2 ≅ R’s, then isos.
11) 1. Given
2. RF ≅ RJ; If isos, then base R’s ≅
4. MF ≅ MJ; If isos, then legs ≅
3. FH ≅ JG; If a seg is added to 2 ≅ seg, then sums ≅
5. OF ≅ KJ; If big seg ≅, then like div ≅
6. ΔOFH ≅ ΔKJG; SAS
7. OH ≅ KG; CPCTC
12) 1. Given 2. FCDG is a trap; If 1 pr opp sides ║, then trap
3. R1 suppRFCD & R2 supp RGDC; If 2 R’s form str R’s (AFD), then supp.
4. RFCD ≅ RGDC; If 2 R’s supp to ≅ R’s, then ≅ 5. FCDG is an isos trap; If trap has lower R’s ≅, then isos.
13) 1. Given
2. ΔKOR is =angular; If Δ=lateral, then =angular
4. RM ≅ RKRO & RP ≅ RRKO; If ║ogram, then opp sides ≅
6. RJKR ≅ RM & RJKR ≅ RP; If ^ lines, then corr R’s ≅
8. KR ≅ KR; Reflexive
9. ΔKJR ≅ ΔROK; ASA
12. ΔJMP is =angular; If all R’s ≅, then =angular
14) 1. Given
5. KR ║ MO & KR ║ OP; If ║ogram, then opp sides ║
7. RJKR ≅ RRKO & RJRK ≅ RORK; Transitive
10. RKJR ≅ RROK; CPCTC
11. RJ ≅ RM ≅ RP; Transitive
13. ΔJMP is =lateral, If Δ =angular, then also =lateral.
2. RBAC & RDCA are Rt R’s; If ^, then Rt.
5. ΔBAC ≅ ΔDCA; SAS
15) 1. Given
3. RRKO ≅ RKRO ≅ RKOR; If =angular, then all R’s ≅
3. RBAC ≅ RDCA; If Rt R’s then ≅
4. AC ≅ AC; Reflexive
6. RB ≅ RD; CPCTC
2. BC ≅ CE; If bisector, then 2 ≅ seg
3. RBCA ≅ RDCE; Vert R’s ≅
4. ΔBCA ≅ ΔDCE; AAS
5. AB ≅ ED; CPCTC
16) 1. list 2 possibilities
Reflexive
2. D is the midpoint of AC; assume one poss. to be true3. AD ≅ DC; midpoint splits seg into 2 ≅ segs
4. BD ≅ BD;
5. RDBA, RBDC are Rt. Angles, perp. Lines form rt. angles 5. RDBA ≅ RBDC; ≅ right angles 6. ΔABD ≅ ΔCBD; SAS
7. AB ≅ CB;
CPCTC 8. AB not ≅ CB, given 9. Assumption was false, contradiction 10. D is not mdpt of AC, only remaining poss.
17) 1. Given
2. AE ≅ ED; If sides ≅, then R’s ≅
3. RBEA ≅ RCED; Vert R’s ≅ 4. ΔABE ≅ ΔDCE; ASA
8
Part II
1) 145º
2) a) S
e) A
b) N
f) S
c) A
g) S
d) A
3) mRF = 140º
22)
23)
24)
25)
26)
27)
perimeter = 58
4) 110º
5) 64.5º
6) a) RHOM
b) PARA
c) RT Δ
d) SQUARE
7) 131º
8) 105º
9) 110º
10) 60
11) a) 2
b) 3 non-collinear
c) lies on same line
d) lies on same plane
12) a) ray
e) line
b) seg
f) line, ray
c) seg
g) line, angle
d) seg
13) a) 40
c) 48
b) 15
d) 24
14) a) 60º
f) 90º
b) 150º
g) RSQY, RDQT
c) 30º
h) RYQT, RDQS
d) 30º
i) RDQM, RMQT, RMQS
e) 90º
j) QT
uuur
mRBAC = 70º
b) mRBAC = 75º
AB = 8
mRBCA = 60º
mRD = 75º ; sides not possible
30) a) F
b) T
c) F
31) 65º
d) F
e) T
32) R2 ≅ R3
33) p ║ q
34) R16 (alt ext) or R13 (corr)
35) R15 (corr) or R14 (alt int)
38) R4 & R6 or R3 & R5
b) TM
19)
20)
21)
g) equilateral
h) bi-conditional
i) perpendicular bisector
28) a) SSS
d) SAS
b) ASA
e) ASA
c) SAS
f) HL
29) a) mRABC = 55º mRACB = 55º
R3 & R7 or R4 & R8
16) a) RR
18)
e) reflexive
f) R’s
37) R1 & R5 or R2 & R6
c) RA, RC
17)
d) corr R’s / sides ≅
36) R3 & R6 or R4 & R5
15) a) BC & AB
b) RB
c) TR , RM
a) acute, scalene b)
c) eq, eq
d)
e) obtuse, isosceles
a) T
d) F
b) T
e) F
c) T
f) T
a) 90º
b) 15º
117º
36º
4
150º
20º, 70º, 160º
63º 43’ & 153º 43’
FEH, GEH
a) equilateral
b) converse
c) isosceles
right, isosceles
right, scalene
g) F
h) T
i) T
c) 150º
39)
40)
41)
42)
43)
44)
45)
65º
59º
50º, 50º, 85º, 95º, 45º
52.5º, 22.5º
80º
80º
45º
RO & RKLO
46) RN & RMLN
47) 3, 4
48) BDC
49) 22º, 22º, 48º
9
50)
51)
52)
53)
54)
55º, 50º, 55º, 50º, 130º
40º
9, 36º
a) 9
b) FE
c) 40
64) (-½, 6), 2/5
65) m AB = 4/3
m BC = -3/4
m AC = -7
AB ^ BC à B is a rt R
All corner angles are 45º
All angles where diagonals intersect are 90º
55)
F
E
15°
15°
G
75° 75°
75°
75°
15°
15°
56)
All corner angles are 15º and 75º
All angles where diagonals
intersect are 30º and 150º
57)
Corner angles are 15º and 42º
Corner angles are 93º and 30º
All angles where diagonals
intersect are 45º and 135º
H
58)
Lower base angles are 15º and 42º
Upper base angles are 108º and 15º
All angles where diagonals
intersect are 30º and 150º
59) 3, 6, 10, 10
60) 7, 17
61) BC
62) 10, 150º
63) 5, 40, 20
•
•
•
•
•
•
So ΔABC is rt Δ
(17, 24)
24
92 (non-convex)
54
168.75
24
conv à If H-F will win, then it is warm today
inv à If it is not warm today, then H-F will not win
contra à If H-F will not win, then it is not warm today.
73) a) -2/15
b) 5/3
c) -7/12
74) a) 3/4
b) 3
c) -6
75) (-22, 3)
76) AD à undef
BC à undef
AB à 0
DC à 0
AC à -1/5
BD à 1/5
AC and BD not opp reciprocals
77) BC à -3/8
AC à 8/3
BC and AC are opp reciprocals
78) x = 4, y = 5, perimeter of KITE = 110
79) a) -3 b) 21 c) isosceles
80) p Þ: g
66)
67)
68)
69)
70)
71)
72)
Exam Notes
110 minutes
35 multiple choice @ 2pts
12 short answer @ 4pts
choose 2 out of 3 direct proofs @ 8pts each
one indirect proof @ 8pts
=
=
=
=
70pts
48pts
16 pts
8pts
Total = 142pts
10